Zero-divisor graphs of partially ordered sets (Q963831)

Let \(P=(X,\leq)\) be a partially ordered set with least element \(0 \in X\). An element \(x \in X\) is a lower bound of \(S \subseteq X\) if \(x \leq s\) holds for all \(s \in S\). Two vertices \(x, y \in X\) are adjacent in the simple graph \(G(P) = (X,E)\) if \(0\) is the only lower bound of \(\{x,y\}\) in \(P\). If the chromatic number \(\chi(G(P))\) and the clique number \(\omega(G(P))\) are finite, then \(\chi(G(P)) = \omega(G(P)) = k+1\) holds, where \(k\) is the number of minimal prime ideals of \(P\).